Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus
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| Format: | Preprint |
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2024
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| _version_ | 1866913482987798528 |
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| author | Sukochev, Fedor Yang, Fulin Zanin, Dmitriy |
| author_facet | Sukochev, Fedor Yang, Fulin Zanin, Dmitriy |
| contents | Let $E(\mathbb{T}^{d}_θ),F(\mathbb{T}^{d}_θ)$ be two symmetric operator spaces on noncommutative torus $\mathbb{T}^{d}_θ$ corresponding to symmetric function spaces $E,F$ on $(0,1)$. We obtain the Gagliardo--Nirenberg interpolation inequality with respect to $\mathbb{T}^{d}_θ$: if $G=E^{1-\frac{l}{k}}F^{\frac{l}{k}}$ with $ 0\leq l\leq k$ and if the Cesàro operator is bounded on $E$ and $F$, then \begin{align*} \|\nabla^lx\|_{G(\mathbb{T}^{d}_θ)}\leq 2^{3\cdot 2^{k-2}-2}(k+1)^d\|C\|_{E\to E}^{1-\frac{l}{k}}\|C\|_{F\to F}^{\frac{l}{k}}\|x\|_{E(\mathbb{T}^{d}_θ)}^{1-\frac{l}{k}}\|\nabla^kx\|_{F(\mathbb{T}^{d}_θ)}^{\frac{l}{k}},\; x\in W^{k,1}(\mathbb{T}^{d}_θ), \end{align*} where $W^{k,1}(\mathbb{T}^{d}_θ)$ is the Sobolev space on $\mathbb{T}^{d}_θ$ of order $k\in\mathbb{N}$. Our method is different from the previous settings, which is of interest in its own right. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_13094 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus Sukochev, Fedor Yang, Fulin Zanin, Dmitriy Functional Analysis 35A23, 46L52, 58B34, 46E30 Let $E(\mathbb{T}^{d}_θ),F(\mathbb{T}^{d}_θ)$ be two symmetric operator spaces on noncommutative torus $\mathbb{T}^{d}_θ$ corresponding to symmetric function spaces $E,F$ on $(0,1)$. We obtain the Gagliardo--Nirenberg interpolation inequality with respect to $\mathbb{T}^{d}_θ$: if $G=E^{1-\frac{l}{k}}F^{\frac{l}{k}}$ with $ 0\leq l\leq k$ and if the Cesàro operator is bounded on $E$ and $F$, then \begin{align*} \|\nabla^lx\|_{G(\mathbb{T}^{d}_θ)}\leq 2^{3\cdot 2^{k-2}-2}(k+1)^d\|C\|_{E\to E}^{1-\frac{l}{k}}\|C\|_{F\to F}^{\frac{l}{k}}\|x\|_{E(\mathbb{T}^{d}_θ)}^{1-\frac{l}{k}}\|\nabla^kx\|_{F(\mathbb{T}^{d}_θ)}^{\frac{l}{k}},\; x\in W^{k,1}(\mathbb{T}^{d}_θ), \end{align*} where $W^{k,1}(\mathbb{T}^{d}_θ)$ is the Sobolev space on $\mathbb{T}^{d}_θ$ of order $k\in\mathbb{N}$. Our method is different from the previous settings, which is of interest in its own right. |
| title | Gagliardo-Nirenberg interpolation inequality for symmetric spaces on Noncommutative torus |
| topic | Functional Analysis 35A23, 46L52, 58B34, 46E30 |
| url | https://arxiv.org/abs/2408.13094 |